Learn Spanish (Spain): Ayer hablé con el empleado. - Yesterday I spoke with the employee.

The accent mark on the final é is what makes it the "I" form in the past tense (preterite). It shows that the action happened in the past, triggered here by ayer (yesterday). The accent is crucial because it changes the pronunciation and meaning. Notice that the "he/she/you formal" past tense form also gets a final accent: habló.

Could I use hablo instead of hablé here?

No, because hablo means "I speak" in the present tense. The word ayer tells us the action is completely finished and belongs in the past, so we must use the preterite form hablé.

Do I need to include yo and say Ayer yo hablé?

You can, but it is completely optional. The ending -é in hablé already tells us exactly who did the action ("I"). Spanish speakers typically drop the yo unless they want to emphasize that I spoke with the employee, as opposed to someone else.

"The employee" is a person. Why isn't there a "personal a" before el empleado?

You only use the "personal a" when a person is the direct object of the verb (the one receiving the action directly without any other connecting words). In this sentence, we are using the preposition con (with). When a preposition is already there connecting the verb to the person, you don't add the "personal a".