Learn Spanish (Spain): Mi hermana le dijo al psicólogo que estaba deprimida. - My sister told the psychologist that she was depressed.

This is an example of reported speech. Originally, the sister would have said 'I am depressed' (estoy deprimida, present tense). Because the reporting verb dijo (she said) is in the past, the present tense estoy shifts a step back into the past, becoming the imperfect estaba.

Why do we need le if we already say al psicólogo?

In Spanish, it is very common and usually required to use the indirect object pronoun (le) even when you explicitly name the person receiving the action (al psicólogo). It acts as a helpful marker pointing to who is being told.

Could I use era instead of estaba for 'she was depressed'?

No, you must use a form of estar here. Emotions, moods, and mental states always take estar rather than ser, even if the depression lasts for a long time.

Why does the sentence use al instead of a el?

Whenever the preposition a (to) is directly followed by the masculine singular article el (the), they must combine into the contraction al. You cannot leave them as two separate words in Spanish.