Paul montre le truc à Marie.

Because montre is the conjugated verb form, while montrer is the infinitive.

• montrer = to show
• il/elle/on montre = he/she/one shows

In Paul montre le truc à Marie, the subject is Paul, so the verb must be conjugated in the third-person singular form: montre.

Le is the definite article, meaning the.

So:

• le truc = the thing / the stuff / the gadget depending on context

French usually needs an article where English sometimes does too, and often where English might leave one out. Here, truc is a masculine singular noun, so it takes le.

Truc is a very common, informal word in French. It often means:

• thing
• stuff
• gadget
• what’s-it
• thingamajig

It is a casual word people use when they do not want to name something precisely, do not know the exact word, or want to sound informal.

So le truc is roughly the thing.

Here, à Marie marks the indirect object: the person to whom something is shown.

Compare:

• Paul montre le truc = Paul shows the thing
• Paul montre le truc à Marie = Paul shows the thing to Marie

With montrer, the thing being shown is the direct object, and the person receiving it is often introduced by à.

So:

• le truc = what is shown
• à Marie = to Marie

Because Marie is a proper name. In French, proper names normally do not take an article in this kind of sentence.

So you say:

• à Marie = to Marie

not normally:

• à la Marie

That would mean something different or sound wrong in standard usage.

The basic order is:

Subject + Verb + Direct Object + Indirect Object

So here:

• Paul = subject
• montre = verb
• le truc = direct object
• à Marie = indirect object

This is very similar to one possible English structure:

• Paul shows the thing to Marie

French also commonly uses this pattern with many verbs.

Yes, in meaning, absolutely.

English has two common patterns:

• Paul shows the thing to Marie
• Paul shows Marie the thing

French usually uses the version with à:

• Paul montre le truc à Marie

So even if the meaning matches Paul shows Marie the thing, the French structure is closer to Paul shows the thing to Marie.

Yes, it is pronounced the same way in standard French.

So:

• il montre = he shows
• une montre = a watch

They are spelled the same in the form montre and are pronounced the same, but the meaning is clear from context.

In Paul montre le truc à Marie, montre is clearly the verb because it comes after the subject Paul.

Yes. The indirect object à Marie can become lui.

So:

• Paul montre le truc à Marie.
• Paul lui montre le truc.

Here, lui means to her.

Notice that in French, object pronouns usually go before the verb:

• Paul lui montre le truc.

not:

• Paul montre lui le truc.

Yes. Le truc can be replaced by the direct object pronoun le.

So:

• Paul montre le truc à Marie.
• Paul le montre à Marie.

And if you replace both objects, you get:

• Paul le lui montre.

That means Paul shows it to her.

This is a very useful pattern to learn:

• le = it
• lui = to her / to him

Truc is masculine.

You can tell here because it uses le:

• le truc

If it were feminine, you would expect la instead.

Learning the article together with the noun is the best habit:

• le truc, not just truc

Yes, but the meaning changes slightly.

• le truc = the thing → a specific thing, already known or identifiable
• un truc = a thing → a non-specific thing, or something introduced for the first time

So:

• Paul montre le truc à Marie = Paul shows the thing to Marie
• Paul montre un truc à Marie = Paul shows a thing to Marie

Both are correct; they just do not mean exactly the same thing.