Learn Russian: Ей грустно, потому что холодно.

In Russian, a person's physical or emotional state is often expressed using an impersonal construction. The person experiencing the feeling is put in the Dative case. So instead of the Nominative Она (she), we use the Dative form Ей (to her). Literally, it translates closer to "To her, it is sad."

Why do грустно and холодно end in -о?

In these impersonal sentences, there is no grammatical noun subject (like "she" or "the weather") doing an action. Because of this, Russian uses predicative adverbs, which usually end in -о, rather than adjectives like грустная (sad) or холодная (cold).

Why is there no word for 'it is' before холодно?

English requires a "dummy" subject like "it" in phrases like "it is cold" or "it is raining." Russian doesn't use these at all. To say "it is cold," you just use the adverb холодно completely on its own.

Does холодно mean that the weather is cold, or that she feels cold?

Because there is no pronoun directly in front of холодно, it refers to the general environment or weather ("because it is cold"). If the speaker meant that she was physically feeling cold, they would usually repeat the Dative pronoun to make that clear: потому что ей холодно (because she is cold).