Conceptually it's because it gets much harder to speed something up, but not that much harder to change the direction of its velocity. So for a force with both radial (along velocity) and tangential (perpendicular to velocity) components, the acceleration points more tangentially than the force does.

A really excellent, conceptually clear exposition of relativity is given in chapters 11 and 12 of Morin's Mechanics. The modified version of F=ma is covered in section 12.5.

I would also say that I have been at least somewhat successful in teaching people about relativity by ttuncating the Lorentz transform to first order in β, as you can prove that you get the full results back from this as the matrix limit

                   N
    lim   [1  -φ/Ν]^
    N→∞   [-φ/Ν  1]

... basically the physical content of Galileo’s transform is that as you accelerate forwards things start moving backwards. The physical content of Lorentz’s transform adds just one thing to first order: that as you are accelerating, clocks ahead of you appear to tick faster, and clocks behind you slower, proportional to their distance in the direction you are accelerating.

That key basic insight is not just the resolution to the twin paradox but the basis for event horizons near black holes and such. (There is a surface at coordinate -c²/a where time stands still.)

But note that the 4-force is parallel to the 4-acceleration. However, its spatial component only agrees with classical acceleration in the comoving frame. In general, there's an additional contribution that's however smaller by a factor of (v/c)^2.