What is meant by "assume P"?

The following considerations will answer your questions.

• The proposition is the foundational element in logics. A proposition is normally the starting point for any propositional logics procedure. Propositions are true or false. Nevertheless, there's no formal definition of the expression "assume" in logics, since it is equivalent to "suppose", "think", "imagine", etc. In all such cases, a proposition is stated, in order to proceed to the calculus. The expression used to introduce it ("assume that", "let's say"..., etc.) is not of importance.
• Logical propositions don't need to correspond to reality. "Assume 3 is a negative number" is a valid logical proposition. If you ask "will the operation (3+3+3)/3 be also negative?", the logical process is sound (and the result is true). Although it does not correspond to the reality we normally experience.
• Assume P effectively means that P is true, and it is the starting point of the process.

When someone begins a proof with a statement 'Assume P', what they are really doing is, in effect, creating within their imagination an artificial world, in which everything about the real world holds, plus also 'P' holds. IMPORTANT: This is generally (I am tempted to say always) done in situations where the prover -does not know- whether 'P' really holds or not (i.e the prover generally doesn't start by assuming things like '3 is a negative number' are true).

Anyway, having created this imaginary world, the prover then performs derivations within that world, eventually arriving at 'Q'. So the truth of 'Q' necessarily follows from the truth of 'P' (if it is true). The prover then takes a giant step: From the fact that 'Q' follows from 'P' inside this imaginary world, the prover concludes that the compound statement 'P->Q' must hold unconditionally in the real world! And it is this compound statement which the prover is really after!

Once the prover has arrived at 'P->Q', this statement may then be applied in other contexts where it is known that 'P' actually does hold; the prover may then conclude that in these contexts, 'Q' actually does hold also.

In Natural Deduction, if a conditional statement of the form P=>Q (that is "P implies Q") may be (syntactically) proven, then this it usually done by demonstrating that: Q can indeed be derived under the assumption of P.

"Assume, for the sake of argument, that P is true.   Under that assumption, Q may be derived using such-and-such valid inferences.   Therefore proving that P=>Q."

|  |_ P      Assumed
|  |  :      Some valid inferences
|  |  Q      Derived somehow.
|  P => Q    Deduced (via rule of '=> Introduction')

If P then Q means that if, hypothetically, P were true, then that would imply that Q is also true. It doesn't matter whether or not P is true. We're trying to prove that if P is indeed true, then it will result in Q being true as well.