Learn French: Il paraît qu'ils ont eu un problème. - It seems that they had a problem.

In French, the conjunction que (that) must drop its -e and add an apostrophe when the next word starts with a vowel. Since ils starts with an i, que ils becomes qu'ils. This contraction is mandatory!

Can I leave out the word qu' here, just like we sometimes drop 'that' in English?

No, you can never skip que (or qu') in French. In English, you can say 'It seems they had a problem' or 'It seems THAT they had a problem.' In French, you must always use que to connect the two parts of the sentence.

Who does the il refer to in il paraît?

Here, il doesn't mean 'he'. Just like the English phrase 'it seems' or 'it is raining', French uses an impersonal il to mean 'it'. So il paraît simply translates to 'it seems' or 'it appears'.

What tense is ont eu and why are there two forms of 'to have'?

This is the passé composé (past tense) of avoir (to have). It is made of the helping verb ont (they have) and the irregular past participle eu (had). Together, ils ont eu means 'they had' or 'they have had'.