How do I know that an acid and conjugate base are present in equimolar amounts?
I'm really confused here!


No, you are not! You only (temporarily) mixed up two things, both of with you have already understood separately.

$$\ce{HA + H2O &lt;=&gt; H3O+ + A-}$$

1. Position of the equilibrium

This is described by the equilibrium constant $K$, which is the quotient of the rate constants for the forward and the back reaction: $K = \frac{k_{forward}}{k_{back}}$. For strong acids, of which you know that they completely dissociate, the equilibrium is on the product side.

2. Equimolarity

There's nothing out of nothing! If one molecule of $\ce{HA}$ dissociates is water, there exactly one molecule $\ce{H3O+}$ and exactly one molecule of $\ce{A-}$ generated. Not more, not less. 

Q: How could another molecule of $\ce{A-}$ appear out of nothing?

A: It can't and it doesn't happen - not even in the LEGO universe.



It is just a proton transfer reaction from $\ce{HA}$ to $\ce{H2O}$; the overall number of atoms does not change!

If is however possible that some $\ce{A-}$ may come from a different source, just by adding a salt $\ce{MA}$ of the conjugate base (think sodium acetylsalicylate) to the solution of your acid. jerepierre has mentioned this in his comment! But for the moment, this is another story.