A lot of strength gains (at least initially) come from improved muscle quality, rather than increased muscle mass. Specifically, the nerves controlling muscle fibers 'learn' how to better co-ordinate themselves and become more efficient. With pushups for example, there are several muscles involved (triceps, shoulder, chest), and the nerves 'learn' exactly how many muscle fibers are needed for the movement, and which order they need to fire, so there is less wasted effort. This is known as "neuro-muscular adaptation".
I wouldn't quite say you "can't" grow muscle while burning fat, but it is difficult, and any gains will be small.
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.
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