Why infinitive after 'Despues de, Antes de'

Buenas noches! I think you want to read the reference section on verbs (infinitives and gerunds). Using the gerund after a preposition as you illustrate is a rule of English grammar; Spanish does this with the infinitive.

In Spanish, gerunds often take the form of the infinitve. Adding "ando" or "iendo" onto the end of a verb is for the present progressive tense, not as a gerund. Adding "ando" onto the end of cenar would make no sense to a native speaker. It would then have to be "está cenando" which in turn would also make no sense, since it is after "después de", which you always put infinitives after. It´s a rather simple rule so it should come to you in time. The same applies with "Antés de" as well.

Because they are two different languages.

Because English and Spanish are two entirely different entities with different rules. There are times--plenty of times--in which translations will not fit snugly, with perfectly aligned word-by-word constructions. That's just not the way it works.

Well you put an infinitive because of the letters used in the Spanish word that is used.

• hablar (to speak) — hablando (speaking)
  beber (to drink) — bebiendo (drinking) vivir (to live) — viviendo (living) The Spanish present participle of regular verbs is formed by removing the -ar ending and replacing it with -ando, or by removing the -er or -ir ending and replacing it with -iendo. The verbs that have irregular present participles still use the same -ando and -iendo endings, but they have changes in the stems. Examples: present participle of venir (to come) is viniendo (coming), and the present participle of decir (to say) is diciendo (saying). To prevent awkward spellings, a few verbs use a -yendo ending in the participle instead of -iendo. For example, the present participle of leer (to read) is leyendo (reading).
  • yo — Estoy escribiendo. — I am writing.
  • tú — Estás escribiendo. — You are writing.
  • él, ella, usted — Está escribiendo. — He/she/you is/is/are writing.
  • nosotros, nosotras — Estamos escribiendo. — We are writing.
  • Bosotros, vosotras — Estáis escribiendo. — You are writing.
  • ellos, ellas, ustedes — Están escribiendo. — They/you are writing. I hope you get it now. 0.0 Good luck Hope this helps