Learn Spanish (Latin America): Él piensa empezar un proyecto excelente.

The verb pensar (to think/plan) is a stem-changing verb. In the present tense, the e in the middle of the word changes to ie for most forms, including él (he). That is why it becomes piensa.

Since empezar is also a stem-changing verb, why isn't it empieza here?

You only conjugate the first verb in a chain. Since piensa is already conjugated to match él, the second verb, empezar (to start), stays in its original dictionary form (the infinitive). Stem changes only happen when you conjugate the verb!

I thought pensar meant to think. Why is it translated as plans to here?

You are right that pensar normally means to think. However, when you use pensar followed directly by an infinitive verb (like empezar), it creates a special phrase that means to plan to or to intend to do something. So, piensa empezar translates to he plans to start.

Why doesn't excelente end in o to match the masculine word proyecto?