Short answer. In one of three ways named by the law: the record of birth appearing in the civil register, an admission in a public document, or a private handwritten instrument. Each is an express act by the father himself, and each carries the same consequences once made.
What the law says
their filiation has been expressly recognized by the father through the record of birth appearing in the civil register, or when an admission in a public document or private handwritten instrument is made by the father
RA 9255, Section 1 — Surname Of Illegitimate Children. Read the full provision →
What the law says
The legitime of each illegitimate child shall consist of one-half of the legitime of a legitimate child.
RA 9255, Section 1 — Surname Of Illegitimate Children. Read the full provision →
Three routes, one requirement
Recognition is established where their filiation has been expressly recognized by the father through the record of birth appearing in the civil register, or when an admission in a public document or private handwritten instrument is made by the father. The common thread is that the father must do it, expressly. Treating the child as his own, paying for school, appearing at family occasions, being named by everyone as the father — none of that is what this sentence asks for. It asks for a document that says so, executed by him.
What each route looks like in practice
The record of birth is the cleanest and the cheapest, because it happens once at registration and needs nothing afterwards; the civil registrar's implementing rules prescribe the form the acknowledgment takes and what the office requires to enter it. A public document is one executed with the formalities that make it public — most commonly a notarised instrument. A private handwritten instrument is exactly that: written and signed by the father in his own hand, which is why questions about it usually turn on proving the handwriting is his.
What recognition establishes
The surname is the consequence people ask about, but it is the smallest of them. Recognition fixes filiation, and filiation is what carries the child's right to support and his share in the father's estate: The legitime of each illegitimate child shall consist of one-half of the legitime of a legitimate child. Those rights follow from the relationship rather than from the name, so a recognised child who continues to use his mother's surname loses none of them. Recognition and the choice of surname are separate decisions, and it is worth keeping them separate when advising on either.
It is not final, and it is not legitimation
The law preserves a way back: Provided, the father has the right to institute an action before the regular courts to prove non-filiation during his lifetime. The right belongs to him alone and ends at his death, so a recognition unchallenged in his lifetime is far harder to unpick afterwards. Note too what recognition is not. It does not make the child legitimate, it does not alter parental authority — which remains with the mother — and it is a different thing from legitimation, which depends on the parents' subsequent marriage.