Why do we equate an indefinite integral to a specific value?

In physics we frequently leave off the limits of the integral when the limits can be figured out from the context. So, in the first case, the actual relation is: $$x(t) = x(0) + \int_0^t \dot{x} \operatorname{d}t'.$$

Most often, though, when the limits are left off the implied limits are over all possible values of the dummy variable. For example: $$ Q = \int \rho(\mathbf{x}) \operatorname{d}x^3$$ is understood to be the integral over all of space.

You are being confused by the shortcut that people took who wrote that expression. They mean for the integral to be taken between definite limits.

It would be more proper to say

$$x(t) = x(0) + \int_0^t \dot{x} dt$$

But that gets longwinded. Most people, when seeing the expression as you gave it, will understand it to mean what I wrote. But technically, it's not the same.