What is wrong with each of these sentences? Why?

This company lets people work at home if they wish.
The boys were made to pay for the damage.

Don't ask me why Helen, it's just grammatical

"It's just grammatical" is not much help.

The issue is the use of the infinitive ('to' + a verb)

In the first sentence the 'to' should not be there because we don't use it with 'let' -

This company lets people to work at home if they wish.

In the second sentence there should be a 'to', because we do use it with make/made -

The boys were made to pay for the damage.

That makes perfect sense, and should be easy enough to understand. How long before JTT comes along to muddy the waters?

That's not mud that he uses to make the waters go brown...

Looking back Con I quite agree. Your response however also leaves the q to some extent unanswered

I suppose, Helen, the answer is, that's just the way we do it

OK you goaded me...

1. Some verbs are followed by the to-infinitive:

I decided to go home as soon as possible.
We all wanted to have more English classes.

Verbs that are used with 'to' + the infinitive include:

Verbs of thinking and feeling:

choose
decide
expect
forget
hate
hope
intend
learn
like
love
mean
plan
prefer
remember
would like
would love

Verbs of saying:

agree
promise
refuse

Other common verbs are:

arrange
attempt
fail
help
manage
tend
try
want

2. Some verbs just use the infinitive without a preceding 'to'; these include some verbs of compulsion, perception, permission, assistance, etc:

make
let
help
see
hear
feel
watch

I made him run to the house.
She let me see her drawing.
We watched the fire burn.

(I will try to think of some more...)

Wow thanks Con, I'm now duly impressed, hope Helen also

The bottom line is there isn't really a 'rule', you just have to notice or learn which verbs take 'to' and which don't.

Yea, Con, just as I had supposed, it's just the way we do it