What is actually being doubled when someone has to "double back"?

You double the distance because you made a mistake to go that way at all. Had you avoided your mistake, you would have saved yourself "2x" the travel (out, and back).

Doubled can mean bent back upon itself. When it is used in this sense, it is often paired with up, down, back, and backward. Hence when your path turns back upon itself you can say that you double back. For examples, see this excerpt from A New English Dictionary on Historical Principles:

8. To bend (a piece of cloth, paper, etc.) over, so as to bring the two parts into contact parallel; to fold; to bend (the body, etc.) so as to bring distant parts into proximity; to close, clench (the hand or fist). Often with up.
_{^{(In quot. 1589, to close (the ears).) c 1430 Two Cookery Bks. 39 Take a pese of fayre Canneuas, and doble it. 1589 PUTTENHAM Eng. Poesie III. xxiii. (Arb.) 282 To solace your eares with pretie conceits after a sort of long scholasticall preceptes which may happen haue doubled them. 1665 HOOKE Microgr. 9 They double all the Stuff .. that is, they crease it just through the middle .. placing the two edges, or selvages just upon one another. 1694 DRYDEN Love Triumph. III. i, The page is doubled down. 1778 MAD. D’ARBLAY Diary 3 Aug., He doubled his fist at me. 1874 BLACKIE Self-Cult. 42 Bending his back, and doubling his chest. 1885 BIBLE (R. V.) Exod. xxvi. 9 Thou .. shalt double over the sixth curtain in the forefront of the tent. 1893 A. H. S. LANDOR Hairy Ainu 54 Crouched as she was, doubled up, with her head on her knees.}}
b. To double up (a person): to make to bend or stoop, as by a blow; hence fig. to finish up, cause to ‘collapse’. (slang or colloq.)
_{^{1814 Sporting Mag. XLIV. 278 Planting a blow on the side of Perrot, which doubled him up. 1883 J. PARKER Tyne Ch. 108 Never saw a man so doubled up [in argument]. 1891 E. W. GOSSE Gossip in Library xxi. 275 This master of science [pugilism], who doubled up an opponent as if he were plucking a flower.}}
c. intr. (for refl.) To become folded together or bent over; to fold, bend.
_{^{? 1650 Don Bellianis 164 With such terrible incounters that the knight .. doubled backward upon his horse. 1875 DARWIN Insectiv. Plants vii. 163 After 10 hrs. 15 m. .. the blade quite doubled up. Mod. His knees doubled up under him. The leaf has been folded, and tends to double over.}}
d. Billiards. (a.) intr. Of a ball. To rebound. (b.) trans. To cause (a ball) to rebound: cf. D_^OUBLET 7.
_{^{1885 Billiards simplified (1889) 50 If you .. hit the red nearly full, so that it doubles down the table [etc.] Mod. You can double the ball into the middle pocket.}}
9. Naut. (trans.) To sail or pass round or to the other side of (a cape or point), so that the ship’s course is, as it were, doubled or bent upon itself.
_{^{1548 HALL Chron., Hen. VIII. II b, If you wil bring your shippe into the bay of Hardines, you must double ye poynt of Gentilnes. 1585 T. WASHINGTON tr. Nicholay’s *Voy*. I. X. 12 B, Having doubled the cape, we passed along. 1665 Phil. Trans. I. 42 To go into the East Indies without doubling the Cape of Good Hope. 1867 FREEMAN Norm. Conq. (1876) I. v. 295 The invaders doubled the Land’s End and ravaged Cornwall.}}
b. intr. To get round. To double upon (in naval warfare): to get round to the other side of (an enemy’s fleet), so as to inclose it between two fires.
_{^{1769 FALCONER Dict. Marine (1789) A a ij b, The lee-line .. cannot so easily double upon the van .. of the enemy. 1856 Emerson Eng. Traits v. 91 Nelson’s feat of ‘doubling’, or stationing his ships one on the outer bow and another on the outer quarter of each of the enemy’s. 1867 SMYTH Sailor’s Word-bk., Doubling upon .. a hostile fleet .. as Nelson did at the Nile. 1875 F. HALL in Lippincott’s Mag. XVI. 751/2 I doubled nimbly round a couple of corners, and paused again}}
10. intr. To turn sharply and suddenly in running, as a hunted hare; to turn back on one’s course; to pursue a winding or tortuous course.
_{^{1596 DRAYTON Legends ii. 382 To the Covert doth himselfe betake Doubling, and creepes from Brake againe to Brake. 1690 DRYDEN Amphitryon IV. Wks. 1884 VIII. 75 See how he doubles, like a hunted hare. 1724 DE FOE Mem. Cavalier (1840) 95 He found the river fetching a long reach, double short upon itself. 1828 D’ISRAELI Chas. I, I. iv. 87 The nogociation doubled through all the bland windings of concession and conciliation. 1864 D. G. MITCHELL Sev. Stor. 306 They suddenly turned to double upon their walk again.}}
11. fig. (intr.) To make evasive turns or shifts; to use duplicity, act deceitfully. ? Obs.
_{^{1530 PALSGR. 525 2, I double, I varye in tellying of my tale. . . Nay, and you double ones, I have done with you. 1578 HUNNIS Hyveful Hunney Gen. xii. 25 Why has thou dealt thus craftely And doubled so with mee>? 1624 Trag. Nero III. iii. in Bullen O. Pl. I. 54 Why with false Auguries have we bin deceiv’d? What, can Celestiall Godheads double too? 1649 Bounds Publ. Obed. (1650) 35 Who have been .. attent not to double with their God. 1820 SCOTT Ivanhoe xxxv, If thy tongue doubles with me, I will have it torn from thy misbelieving jaws.}}

I have always seen the phrase used as a tactic to throw off pursuit. As in looping wide around in a circle on a trail until you have come across your original tracks. In so doing you are now behind the party who is following you. Which now gives you the advantage to sneak up on them while they think they are sneaking up on you.

There is another as well.

When being pursued you go one direction and then do the same as before, loop around wide back to the original trail and then go a different direction once you meet the beginning of the first loop. In so doing you make two trails for the pursuing party to decide over following once they reach the split. The advantage in this is to

a) Throw the party off entirely, if they are to chose to follow the wrong path. making them loop back around, eventually meeting up with the original trail, putting distance between you.

b) Having the party split in half, one half following the false trail the other half following the correct trail. In so doing their force is only half as strong.

To double back, is to go back out of your way in the opposite direction you are heading. The distance between the point where you are at the moment you decide to turn back and go to a previous point where you were is arbitrary. But going back to that previous point and then returning to where you are at the moment you choose to double back is twice the arbitrary distance. So in order to double back and then continue on your original path, represents covering the distance twice. It can be a literal or figurative or metephorical application. For example:

You left your house for the Airport that is 30 miles away. After you traveled 10 miles from your house, you realize you left your passport, that you need to take with you on your flight to travel. So you turn around and double back 20 miles to your house to get your passport. 10 miles back to your house and 10 miles back to the point you were at when you realized you had to double back. When you miss your flight and you have to re book the next flight, you might tell someone who needs to know that the reason you missed your flight is because you had to go out of your way to double back home to get your passport when you were half way to the Airport. So in simple terms the meaning of double back is actually to go out of your way for something or some reason.

Another example that is metaphorical: “Mark is a good guy, he will always double back for a friend.” This a way of saying that Mark is a good friend, he will always go out of his way to help a friend in need.

Double back: to go out of your way from your path in order to fulfill an objective before returning to your path.