As rspreng said, it just has to be there. There are many verbs in Spanish (and English) that require that you add a preposition to the end.
Empezar (a) = To start/begin
Comenzar (a) = To start/begin
Tratar (de) = To try ("tratar" does have other meaning, such as "to treat")
Aprender (a) = To learn (And as AndresC. said, you only add the "a" if it's followed by a verb)
Some examples of this happening in English are:
To look (at) = Mirar
To listen (to) = Escuchar
Notice that "mirar" and "escuchar" don't require a preposition, but the English equivalents do. There isn't really a reason to all this, it's just how it's done.
Note: If you wanted to add an object pronoun, you don't need the preposition. So you say "Aprenderlo" not "Aprender a lo" or "aprender a ello."
Aprender does not require "a" in all situations, por ejemplo:
-Quiero aprender a hablar alemán
-Quiero aprender alemán
As you see, aprender+verb requires the preposition, whereas aprender+noun doesn't.
The verb "mirar" does use the preposition "a" in most of its common uses.
-El hombre mira a su esposa
-La casa mira al mar
-¡Mírate al espejo!
And the same goes with "escuchar". In many cases, you need the "a".
-Escucha a esa persona, las cosas que está diciendo de tí.
(Listen to that person, the things he's saying about you).
Well, going back to the sentence that prompted the question in the first place, the rule is very simple: in Spanish you don't put two non conjugated verbs next to each other. The preposition will change accordingly, but there will always be one of them in the middle.
Empezar a comer
Volver a viajar
Comenzar a estudiar
Terminar de cocinar
Pensar en regresar
Tener que trabajar
Estudiar para progresar
Hope it helps.
Look (at) is a bad example because it's a phrasal verb and 'at' is used to introduce the indirect object, it isn't only used to catenate to another verb. It is important to differentiate between prepositions as part of phrasal verbs, and 'prepositions' only used between verbs; I use the quotes because in English we refer to the use of 'to' between verbs as a particle, not a preposition.
Anyway this sentence should be relatively straightforward because similar to empezar and comenzar, start and begin also use the 'to' particle when catenating.
Here's a page that has every verb and what preposition it needs: http://www.elearnspanishlanguage.com/grammar/verb/verbswithprep.html
I didn't find a huge number of examples but I think that comiendo should be possible although the meaning changes slightly:
"Según la experta, si empezamos comiendo alimentos poco calóricos muy despacio, podemos sentir antes la sensación de saciedad y comeremos menos de otros alimentos que aportan más calorías."
“La cocina, que también tiene su parte estética porque empezamos comiendo por los ojos, y la escultura, coger una piedra y hacer algo que visualmente te alegra el corazón y el alma.”
"Nosotros vamos a empezar comiendo" implies that you will begin by eating, then..... something else happens. It implies a progression of some sort.
On the other hand, "Nosotros vamos a comer." Punto final. We start to eat. Then that's it. Nothing more.
It's just the way Spanish works. There are a handful of verbs that require the gerund afterward, generally they don't use a. So you can continuas comiendo but not empieza comiendo. And please, stop translating word for word. It only confuses you. Spanish just uses the infinitive in places where English uses the gerund.