Why is F $\implies$ T taken as true? Why is this the "convention"?
It's a convention in classical logic. Not all logical systems make this convention.
To see why this convention makes sense, consider a statement of the form $P\implies Q$, which I make, swearing that I'm telling the truth. When will you be able to sue me for lying? Well, if $P$ is true, but $Q$ is false then certainly you can sue me. But if $P$ is not true and $Q$ is, and you try to sue me, the judge will wonder what the hell you want. I only said that if $P$ is true then $Q$ is true. So, if $P$ is false, I did not commit to anything and thus your case is dismissed (i.e., I was telling the truth).
The point is that the implication is material implication, not that of causal implication. You may disagree with that interpretation and develop a logic where it is interpreted differently (for instance, in paraconsistent logic a statement of the form $F\implies T$ is dismissed as meaningless, not having any truth value (simply saying, don't bother about such nonsense as pondering what results from a false assumption)). Still, using classical logic in mathematics is very powerful, and so we make this convention.