Learn Spanish (Spain): Mi amiga me dijo que pensaba empezar la terapia. - My friend told me that she planned to start the therapy.

This is the core rule of reported speech! Because the reporting verb (dijo) is in the past, the tense of the original quote shifts back. The friend originally said Pienso empezar... (I plan to start...). When reporting this in the past, the present tense pienso shifts to the imperfect pensaba.

Doesn't pensar usually mean "to think"? Why does the translation say "planned"?

Yes, pensar means "to think", but when it is followed directly by an infinitive verb like empezar, the phrase means "to plan" or "to intend" to do something. So, pensaba empezar translates to "she planned to start".

Why doesn't dijo que trigger the subjunctive in the second part of the sentence?

The verb decir (to say/tell) only triggers the subjunctive when it is used to give a command or influence someone, like if she told you to start therapy. Here, she is just reporting a fact or sharing a statement of her own intent, so we use the standard indicative mood (pensaba).

Could we use the preterite pensó instead of the imperfect pensaba here?

No, in reported speech, a present tense original quote always shifts to the imperfect. Using pensó would sound unnatural here because it implies she had a sudden, completed thought at a specific moment, rather than describing the ongoing intention she had at the time.