We use has, had, or have with a past participle to form the present perfect. This contrasts with our use of a single verb for the present simple. We do not say “I taken the test,” but instead use the simple past tense took and say “I took the test.” For what grammatical reason do we use have, had, or has with a past participle?
How the have-perfect came to be
The OED has this to say about the verb to have with regard to its use as an auxiliary for creating perfect tenses when combined with some inflection of the verb have the way that you’ve asked about:
IV. As an auxiliary verb, used with the past participle of another verb to form the perfect.
The have-perfect in English apparently arose as a reanalysis of uses such as I have my work done ‘I have my work in a done or finished condition’ (see sense 7b); the complement done was reinterpreted as part of the verb phrase, a process which was reinforced by a lack of fixed word order and the possible transposition of object and participle, i.e. I have done my work. This development appears to have largely taken place before the written record. Even in early Old English, in the majority of examples with transitive verbs the past participle is not inflected to agree with the object. Despite occasional ambiguity, there are few Old English examples in which the past participle must be regarded as a complement rather than as part of a perfect construction.
In Old English, the have-perfect is not only established with transitive verbs, but also with intransitive verbs expressing action or occurrence, while the perfect of intransitive verbs expressing change of state or position is usually formed with be (be v. 16b). From Middle English onwards the perfect with have gradually becomes more common in these verbs, and is the predominant form by the early 19th cent., except in contexts where the focus is on resultant state (for example, she is gone is still typically used to express state, while she has gone expresses action; such usage is now, however, quite limited). In early Middle English the have-perfect also extends to verbs denoting ongoing states or conditions, and to the verb to be. Compare discussion in etymology section.
Sense 16b of be covers how be was also historically used as an auxiliary with a past participle to form perfect tenses of intransitive verbs:
b. In intransitive verbs, forming perfect tenses; in later use chiefly with verbs of motion such as come, go, rise, set, fall, arrive, depart, grow, etc., expressing a condition or state attained at the time of speaking, rather than the action of reaching it, e.g. ‘the sun is set’, ‘our guests are gone’, ‘Babylon is fallen’, ‘the children are all grown up’.
Now largely replaced by have following the pattern of transitive verbs: see have v. VI.
An example in (Early) Modern English of using be for a perfect which they provide is from Shakespeare’s Comedy of Errors:
These are the parents to these children,
Which accidentally are met together.
Today we would say have met to form the perfect there.
A smidgen of the discussion alluded to in the etymology section for have is:
Use as auxiliary.
The development of the periphrastic constructions with past participle (the antecedents of the modern present and past perfect) to some extent parallels developments in other Germanic and Romance languages, but appears at least partly to reflect development within English (compare note at branch VI.). Although frequently attested in similar contexts to the later present and past perfect, in early use the periphrastic constructions are variants of the simple past tense and their use is not fully grammaticalized. With transitive verbs in Old English, they are sometimes not easy to distinguish from originally related uses in other senses where the participial adjective functions as complement (compare not only sense 7b, but also sense 28a).
The OED has more than 50 dense pages of notes on this verb’s historical forms and etymological information before it even starts talking about its myriad senses and subsenses. Remember too that the past participle was used with be not just with have historically, and that be is a far longer entry than have!
Given all that, I really don’t think that we’re going to be able to provide much more than a palimpsest of a full answer here in our format, which does not lend itself to such things.