Transposing instruments cause a lot of grief for music students. In actual fact if the concept is understood It’s not nearly as confusing.
Let’s consider the four families of the orchestra. We have, looking at an orchestral score, from the top down,:

1) The Woodwind Family

Piccolo
Sounds an octave higher than written but this is not transposing. It reads ‘C’ and it sounds ‘C’. If the music notation was written at pitch it would involve numerous leger lines which would make reading all the more difficult.

Flute
This is not transposing. It reads ‘C’ and it sounds ‘C’.

Oboe
This is not a transposing Instrument so no problems here. It sounds as it reads.

Clarinet
There are various different ones and all are transposing. I am considering the B flat Clarinet as an example because it is a popular one.
When a B flat Clarinet reads C, it actually sounds B flat.
This means that it is sounding a tone lower than the written notation.
This does not effect the Clarinetist, but
it does effect the musician in two different ways:
1) When arranging for a Clarinet and maybe a Flute, the musician would need to understand how transposition works.
If for example the musician was writing a piece of music in the key of D Major there is no problem with the Flute.
On the other hand if he writes for the B flat Clarinet in D major he will have a serious problem. This is because, as I explained above, the B flat Clarinet will sound a tone lower so it will sound in the key of C major.
Therefore, in order that it sounds in the key of D, the music must be written a tone higher than the required key so that it sounds a tone lower, in D Major.. What key would the musician need, to sort the problem out?
Answer, the key of E major. So one must write the music a tone higher in order that it sounds in the correct key, here the key of D Major.

One can get a clarinet in C which is not transposing.

Cor Anglais
The name translated means English Horn but it is neither of English origin or a Horn.
The Cor Anglais a transposing instrument and at one pitch only, ‘F’. It is in fact part of the Oboe Family (Oboe, Cor Anglais and Bassoon). It’s notes and finger positions are the exact same as the Oboe but it sounds a 5th lower.
When it reads C, it sounds a perfect 5th lower, that of ‘F’.
This means that in order to sound in the correct key one must transpose the music up a perfect 5th higher than the required key.
Again, consider D Major. What Key would the music need to be written in, to sound in D major?

Answer, key of A major. It will sound a perfect 5th lower which would be D Major.

Bassoon.
This is the lowest of the woodwind family and is not transposing.

Double Bassoon.
The Double Bassoon, sometimes called the contrabassoon sounds an octave lower than the written music. This is to avoid many leger lines and makes for easier reading. The Double Bassoon is not transposing.

Note: Any instrument in ‘C’ is not transposing.

One family of the orchestra is enough to discuss at a time so I’ll leave it at that for now.

I did say that transposing instruments effects the musician in two different ways. I’ll also come back to that later.