When we talk about compilers, we often need to talk about languages. A meta-language is a language you use to talk about another language.
For example, if I say: In Swedish, we say "det är soligt" to say that the weather is sunny.
In this sentence, I'm using English as a meta-language to talk about Swedish, which is in this case the object-language. The language I use is English (the sentence is in English). But the language I talk about is Swedish.
The main cause of confusion when using a meta-language is when you don't know in what language a statement is made. This confusion can often happen when reading Typing Rules.