How do I straighten my bent metal cupboard frame?

Nice cabinet by the way. I'm guessing it's a ferrous metal and if you hold a magnet to it the magnet is attracted to the metal. If so get your self a Mapp gas tank and a swirl-type torch. Heat the bent metal until it turns a dull red. Move the torch across the buckled area keeping the metal hot. Now with a smooth-face hammer (any style, 20-24 oz is best) begin tapping the bent metal back into shape. Start with light blows until you get a sense of how the metal is reacting to the hammer strikes. Re-heat as necessary.

If the metal is solid and not a tube you will need to apply heat for awhile. If you find the metal isn't moving under the hammer strikes heat it to a bright red or orange color. It may be easier to turn the cabinet on it's side or top to facilitate the pounding. Wear heavy gloves, eye protection and have a bucket of water close by.

Assuming that the bent part is actually a square hollow tube and that the foot is open:

• Lay the cupboard on the ground with the bent leg horizontal and on top - i.e., if we're looking at the right side front leg, lay the cupboard either on the back or the left side.
• Get a metal bar slightly smaller than the inside of the leg. A piece of rebar might fit. You want something as strong as possible (without costing a fortune...) and at least a few feet long because this is all about leverage.
• Insert the bar into the leg just up to the bend (i.e., approximately where the crosspiece is welded to the leg).
• Bend hard!
• As you gradually get it bent back into shape, slide the bar farther into the leg and repeat.

You may need some helpers to hold the cupboard steady as you bend the leg, but if you can place one side of the cupboard against a brick wall (or similar) and brace yourself appropriately then you may be able to do this by yourself.

There is good information given on how to bend steel tubing.

The door won't close because the door frame is bent, not just the leg.
Here's a drawing to illustrate the issue.

It is one thing to get the leg straight - it's another thing to get the door to fit in the frame.

You'll need to straighten both the frame and the leg.
Using a square do an analysis of everything that should be square and straight to establish a base line before you start bending. The key to success is knowing what is messed up so that you can bend a little and check to evaluate progress then bend a little more. It may be that the impact to the bottom corner of the cabinet skewed the others corners as well.

Remove the door, if possible (and anything else that gets in the way of working on the cabinet) and then follow the bending instructions given in the other answers. As you bend you'll need to bend both the leg and the frame. Use your square often as you bend so that you check your progress.

Keep an eye on all four corners of the door frame as you bend. You need four 90 degree angles to get the door to shut well.

Metalmongery isn't something I've developed skills for, so my inclination would be to avoid glowing redness, loud clanging, and the likelihood of cracked welds by my hammed fists.

I'd consider making a square cut at the top of the bent portion of the frame and removing that bit. Fit a new section of box tube using an interior insert and metal-impregnated epoxy (JB Weld). You wouldn't want to rely on the epoxy for bending force, so you'd need angle or tubing fit well inside the joint. Ideally you'd connect the horizontal tube that way, too, with a hole into the new leg section. Apply epoxy liberally to all connections and let it cure well, then sand and paint.

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