1

I'd like to know how exactly 'infinitive' modifies a noun.

  1. I suggest you write out the speech you will be giving at the conference and practice it as many times as possible. When you think you're ready, gather up some people to listen to your speech, so you can have a rehearsal before the conference starts.

what's the meaning of people to listen to your speech?

(a) people who is willing to listen to your speech.

(b) people who is able to listen to your speech.

  1. If you need anyone to talk with about this matter, just remember that I'm always here.

what's the meaning of anyone to talk with about this matter?

(c) anyone who is willing to talk with you about this matter.

(d) anyone that you can talk with about this matter.

I don't think there's an answer among abcd, so please let me know know to transform it to a relative clause.

1

Those usages are both definition 2a from Merriam-Webster's online dictionary:

To: 2a —used as a function word to indicate purpose, intention, tendency, result, or end

In both those sentences, the people or person are being described by the purpose that they are going to serve for the listener.


Your first example

  1. When you think you're ready, gather up some people to listen to your speech, so you can have a rehearsal before the conference starts.

Could be rephrased:

When you think you're ready, gather up some people who have listening to your speech as their purpose, so you can have a rehearsal before the conference starts.

or

When you think you're ready, gather up some people who intend to listen to your speech, so you can have a rehearsal before the conference starts.

Your second example

  1. If you need anyone to talk with about this matter, just remember that I'm always here.

Could be rephrased:

If you need anyone whose purpose is talking with you about this matter, just remember that I'm always here.

or

If you need anyone who intends to talk with you about this matter, just remember that I'm always here.


Other examples:

I need someone to mow my lawn.

I'm here to fix the sink.

I use a permanent marker to label my food.

  • That was a night to remember. – Tᴚoɯɐuo Jul 9 '15 at 21:43
1

People that are to listen to your speech. This can be derived from a longer formula: people that are wanted by you to listen to your speech.

1

gather up some people to listen to your speech

This can be taken as an abbreviated form of:

gather up some people (in order for them) to listen to your speech

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