How many Degrees of Freedom do Linear Triatomic Gas Molecules Have and Why?

It is true not only for triatomic but for any molecule.

If your textbook is on thermal physics, then a linear triatomic molecule is said to have $3$(translational)$+2$(rotational)$+3*3-5$(vibrational)=$9$ d.o.f. For all vibrational modes to be active requires high temperatures. So normally at moderate temperatures, only $2$ vibrational modes are active. That's why people talk of $7$ d.o.f for a linear triatomic molecule.

For a linear triatomic molecule, there are 3 degrees of freedom for each molecule, i.e., a total of $3N = 3 times 3 = 9$ degrees of freedom. Since there are 2 constraints (say, bonds) between the 3 molecules, the total degrees of freedom are now $3N - 2 = 7$ degrees of freedom. Out of which 3 are translational, 2 are rotational, and the rest 2 is vibrational.