Learn French: Est-ce que c'est un grand problème ?

The first one is part of est-ce que, the standard phrase used to turn a statement into a yes/no question. The second one is in c'est (it is). Literally, the sentence asks 'Is it that it is a big problem?'.

How would I change this into a 'why' question?

Just put pourquoi (why) right at the beginning! You get Pourquoi est-ce que c'est un grand problème ? (Why is it a big problem?).

Why does grand come before problème? Don't French adjectives usually go after the noun?

Most do! But there is a small group of very common adjectives that go before the noun. They usually describe Beauty, Age, Number, Goodness, or Size (often remembered as the BANGS rule). Grand (big) falls under Size.

Problème ends in an 'e'. Why does it use the masculine un instead of the feminine une?

While many French words ending in 'e' are feminine, problème is a famous exception. It is always masculine, so it must take the masculine word for 'a', which is un.