Are those phrases correct?
No. They are unidiomatic.
Explain what it is to me.
Describe the dish to me.
Are there other verbs that skip the preposition “to”?
Yes. A lot of verbs that take both a direct object and an indirect object, allow the indirect object to appear first, without the preposition “to” (or sometimes “for”). That construct is called the dative alternation.
The term “alternation” suggests that the construct with the direct object first, followed by “to” (or sometimes “for”) + the indirect object, is the “standard” construct.
But when the stress in more on the direct object than on the indirect object, and when the indirect object is short (such as a simple personal pronoun or a name), I would say that the dative alternation constuct is the preferred form.
Show me the money. (also correct: Show the money to me.)
Give her the toy. (also correct: Give the toy to her.)
Lend him some change. (also correct: Lend some change to him.)
Tell us the truth. (also correct: Tell the truth to us.)
Ask me anything. (also correct: Ask anything to me.)
Read them a story. (also correct: Read a story to them.)
Write him a letter. (also correct: Write a letter to him.)
Teach him a lesson. (also correct: Teach a lesson to him.)
Promise her the moon. (also correct: Promise the moon to her.)
Do me a favour. (also correct: Do a favour for me.)
Make me a sandwich. (also correct: Make a sandwich for me.)
Buy her a ring. (also correct: Buy a ring for her.)
The dative alternation constuct does not work with all verbs that take both a direct object and an indirect object. It’s unidiomatic to use it with the verbs announce, attribute, confess, convey, declare, dedicate, deliver, describe, explain, introduce, mention, narrate, present, propose, recommend, refer, return, reveal, say, sell, submit, suggest, transfer, …
(Or with the verbs accustom, answer, ascribe, compare, condemn, confine, contribute, exhibit, liken, …)
People might feel differently about some of these verbs. For instance, Jonathan Swift wrote to rev. dr. Thomas Sheridan:
If your worship will please to explain me this rebus, …
And Jordan Belfort, the Wolf of Wall Street, challenges a few audience members:
Sell me this pen.