For the first condition ( teammatesChampionNames.All... ), the idea is that if all ChampionName s of all Particiants of all Teams are collected, you want the Matches where all of these names occur in teammatesChampionNames . This is true if teammatesChampionNames contains these names and when there are as many of these occurring names as there are in teammatesChampionNames .

Applying the same reasoning to the second condition, the query can be rewritten in this form that EF can translate:

return await
(
    from match in _context.Matches
    let championNames = match.Teams
        .SelectMany(x => x.Participants)
            .Select(p => p.ChampionName)
    let teammates = championNames.Where(n => teammatesChampionNames.Contains(n)).Distinct()
    let enemies = championNames.Where(n => enemiesChampionNames.Contains(n)).Distinct()
    where teammates.Count() = teammatesChampionNames.Count()
       && enemies.Count() = enemiesChampionNames.Count()
).ToListAsync((;