What does it mean to be "independent of machine model"?
The class $\mathsf{P}$ is usually defined as the class of decision problems solvable in polynomial time by (one-tape) Turing machines. However, you get the same class if you replace "one-tape Turing machines" by "multi-tape Turing machines", "RAM machines", "machines running C natively", and many other machines. That's what is meant by the class being independent of the machine model.

Other classes, such as $\mathsf{TIME}(n)$, are very much dependent on the machine model: there are (I think) languages which can be computed in linear time on a two-tape Turing machine but not on a one-tape Turing machine.

The reason we are happy that $\mathsf{P}$ is machine-independent is that it makes $\mathsf{P}$ a very natural class. You don't have to ask yourself whether the correct model is Turing machines with one tape, two tapes, or more: all definitions result in the same class. The same cannot be said about $\mathsf{TIME}(n)$, for example.