When searching for the definition of a multi-word verb, try pre-pending "to"; I got excellent search results when I googled "to do better". This entry in MacMillan was especially interesting in that it lists multiple synonyms of "to do better"; hopefully a learner would be familiar with one or more, which would help to understand the original phrase.
Another solution is to break it down. You're faced with "do better", which is composed of two parts: the verb, "to do", and the adverb describing the verb, "better". "To do" we probably already understand, but to provide a short explanation... "To do [something]" means to perform a given action. "To do" alone describes someone/something's general state of being; ex. "to do well", "to do poorly", and your example "to do better".
If I am doing well, overall I am functioning at an above-average level. If I am doing poorly, I'm functioning at a below-average level. If I am doing better (implied: "than I was before"), I am currently functioning at a higher level than I was at some other point in the past.
So in your example, they're trying to say that his overall performance improved for a while (and I have a feeling the next sentence talks about how his performance ended up dropping again afterward!)