Questions & Answers about Я болею за их команду.
Russian болеть has two common meanings:
To be ill / to have a disease
- Pattern: болеть чем?
- Я болею гриппом. – I have the flu.
- Pattern: болеть чем?
To root for / to be a fan of someone or a team
- Pattern: болеть за кого? / за что?
- Я болею за их команду. – I support their team.
- Pattern: болеть за кого? / за что?
So the preposition за after болею tells you this is the “root for, cheer for” meaning, not the “be ill” meaning. Context and the construction болеть за are the key.