In C/E, the part before the slash names the chord and the part after it names the bass note. C/E is therefore a C chord with E as the lowest sounding pitch.
Why inversions are used
Bass notes can connect chords smoothly. The progression C, G/B, Am creates a descending C–B–A bass line. Even with familiar upper notes, that bass movement changes the arrangement.
Playing slash chords on guitar
You do not always need every theoretical chord tone. Find a voicing where the slash note is the lowest audible pitch while keeping the characteristic notes of the chord.
Transposing slash chords
Both parts move. Raising C/E by two semitones gives D/F#. Transposing only the root would leave the bass relationship incorrect.
When to simplify
You can try the main chord during early practice, but listen to whether the bass line is part of the identity of the passage. Removing an inversion can make a progression feel less connected.
Listen to the bass line separately
An efficient way to hear inversions is to follow only the lowest note at each change. C, G/B, Am and F may form a melodic bass line that explains why slash symbols are used even when ordinary chord shapes could technically work.
In an ensemble, agree on which instrument carries that bass motion. A guitar does not always need to duplicate every inversion if the bass or keyboard already makes the line clear.