What are the steps in the process of validation?

The quote by Leonard Peikoff comes from OPAR, p. 8:

"Validation" I take to be a broader term than "proof," one that subsumes any process of establishing an idea's relationship to reality, whether deductive reasoning, inductive reasoning, or perceptual self-evidence.

The 1976 tape recorded lecture series that preceded and spawned OPAR put it this way (quoted in The Ayn Rand Lexicon under the topic of "Validation"):

"Validation" in the broad sense includes any process of relating mental contents to the facts of reality. Direct perception, the method of validating axioms, is one such process. "Proof" designates another type of validation. Proof is the process of deriving a conclusion logically from antecedent knowledge.

The next paragraph in the question (not part of the quote from OPAR) states: "To those who use a method of validation, no explanation is necessary." This is ambiguous. If it means no explanation of the method is necessary, it's incorrect. One still needs to understand the method one is using and why it can successfully establish an idea's relationship to reality. If the questioner's formulation means no further explanation of the idea being validated is necessary, it depends on how effectively the method has actually established the idea's relationship to reality. My understanding of the OPAR quote is that it is saying mainly that any process of establishing an idea's relationship to reality is a process of validation -- without necessarily implying that any possible process of that kind is assured of being successful and effective.

The context of the OPAR quote is the use of "perceptual self-evidency" to validate the metaphysical axioms. To understand more clearly why direct sense-perception is sufficient for those axiomatic concepts (sufficient to establish the concepts themselves, but not to establish that they are axiomatic, as OPAR explains on p. 11), one must read the entire section of OPAR titled, "Existence, Consciousness, and Identity as the Basic Axioms," starting on page 4. It's also very helpful to have read at least the chapter on axiomatic concepts in ITOE, along with related passages in other chapters.

The question also states: "To those without a method of validation, no explanation seems adequate." Good observation.

The question asks: "In the case of perceptual self-evidence, is validation just a matter of observing a bunch of similar entities such as dogs, triangles, bricks, etc., and associating the objects with the words we use to refer to them?" OPAR does not say that direct sense-perception is sufficient to establish the relationship of any idea to reality. The OPAR discussion mentioned in the question pertains specifically to axiomatic concepts. Additional methodology is needed for non-axiomatic concepts, as described in both OPAR and ITOE. Additional methodology is also needed to establish that a concept is axiomatic (if it is).

It may seem paradoxical by conventional thinking that an "axiom" would require proof that it is axiomatic. Axioms are conventionally regarded as essential preconditions for any proof, and therefore unprovable without circularity (or unprovable tautology). But Objectivism shows that the truth of axioms is demonstrable by sense-perception, and that their status as axiomatic requires further observation and reasoning, which Objectivism provides. Objectivism shows why certain concepts are demonstrable as true (corresponding to reality) only by example and are implicit in all knowledge and thus inescapable.