Szpiro's conjecture is the statement that given $\epsilon$ > 0, there exists a constant $C(\epsilon)$ such that for any elliptic curve $E$ defined over $\mathbb{Q}$ with minimal discriminant $\Delta$ and conductor $f$, we have:
$$|\Delta|\leq C(\epsilon) f^{6+\epsilon}$$
I'm familiar with all the terms, and the wording itself makes sense to me - I just cannot wrap my head around what that formula actually means. Much in the same way continuity can be defined in the epsilon-delta way, or simply "you can draw it without lifting your pencil", is there a less rigidly mathematical way of expressing Szpiro's conjecture (already assuming knowledge of elliptic curves)?
The (modified) Szpiro conjecture is just equivalent to the $abc$-conjecture. Actually, the $abc$-conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves. Here the conductor is a numerical invariant of $E$, which has deep connections to difficult number theory (see for example this question and the related questions there). The $abc$-conjecture basically says, that if you have, for coprime integers $a,b,c$, $$ a+b=c $$ then not all of $a,b,c$ can contain high powers of primes. For example (a toy example), if you have $$ a+b=2^{10}+3^4=1105=5\cdot 13\cdot 17 $$ then $1105$ does not contain "higher" powers of primes.