As a euphonium player I have often wondered, why are treble clef parts written for a Bb insturment, and bass clef parts written for a concert C instrument? I understand why different instruments have transposing parts but have been at a loss fto find out why lower brass parts have this anomaly!
The answer here is simple:
To clear up any misconceptions about low brass: Euphoniums, tubas, and trombones are considered non-C non-transposing instruments. In other words, their fundamental pitch is not "C" (Bb, Eb, F, or C) but in bass clef, they read concert pitch.
Now to answer the question directly:
Baritones / euphoniums are written also in treble clef as a matter of practicality. Middle school age students that switch to low brass are too small to wield even an Eb tuba. As a result, they are taught to play baritone / euphonium instead. Students that play low brass are typically trumpet players initially, so they already understand the basic fingerings for the treble clef.
Playing euphonium in the treble clef is like giving a student a trumpet that plays an octave lower. They can concentrate on learning the instrument without the hassle of trying to master a new clef and new fingerings at the same time.
Eventually, students are encouraged to learn bass clef and move appropriately.
As a corollary, British brass bands typically only read music in treble clef, so there is historical precedent (as well as expectation) there.
Hope that helps.
I don't think there's any reason other than historical chance. Might as well ask why string parts jump from one clef to another (just ask us cellists!). Sometimes it avoids ledger lines, sometimes not; alternatively look at clarinet parts which often go a large number of ledger lines above or below the staff, but never is a different clef used.
Your misconception is about that bass clef being an anomaly, since you're already aware of transposing instruments, apparently.
The clef used for concert pitch notation is mainly dictated by the range of an instrument, the goal being to try to avoid an excessive use of ledger lines. Low brass instruments are (obviously) low range and therefor written in bass clef, high instruments in treble clef. Mid-range instruments can be written in alto or tenor clef.
It's also not unusual for an instrument to use more than one clef, for example high range parts for the tenor trombone have been written in tenor clef. Most modern composers try to avoid exotic clefs, though.
Transposed notation was originally used for instruments, that were tuned to specific scales (mainly clarinets back then). Players had to switch instruments for different scales and the notation allowed them to maintain their finger positions.
The concept was quickly adapted by brass players to allow them to switch freely between instruments and maintain their valve positions for each 'note', thereby only being required to adapt embouchure to the new instrument. It's the default notation style for most traditional brass bands.
(This is where I disagree with jjmusicnotes)
So the reason for transposed notation is not to allow students that started with the trumpet an easier access to low brass instruments. That's just a practical side effect.
A tenorhorn/euphonium player in a traditional brass band will usually not be required to learn bass clef notation, in fact quite a few never do. Pretty much everything they'll ever have to play will is written in transposed notation and even if not, transposing it is a non-issue these days.
The premise of your question is a little off, as not all treble and bass cleff instruments have Bb and C transpositions respectively; however, we can ask why euphonium parts specifically ended up being written with two different transpositions. (As a euphonium player myself, I always found this rather vexing.)
Originally, euphonium music, even in bass clef, was typically written in Bb. I imagine that is due to the fact that the valve system for a euphonium evolved from those of trumpets trumpets; Euphoniums are in Bb because trumpets are in Bb. Why trumpets are in Bb is a longer story...
As for why it eventually became standard for bass clef euphonium music to be written in C instead of Bb... I have to admit that I don't know the answer for sure, but I strongly suspect that this developed in order to allow euphoniums to read off of the same music as other bass clef instruments like trombones.