Explain why the acceleration of the shuttle increases while the rocket's engines are burning fuel

As fuel burns, there's obviously less of it to carry, so it can accelerate faster as it becomes lighter and lighter.

on the shuttle at launch not only does the aceleration increase but also the thrust force itself. the SRB solid boosters used at launch have a thick ring of combustible propelant around a hollow core. a flame is lit in the core which ignites the propellant, only the air facing surface of the propellant burns so over time as the radius of the cavity widens the surface area burning is boosted and hence th e thrust. for a good understanding of how rocjets work try tsiolkovskys rocket eqn.