Target needs to hear you, but not because of how giving inspiration normally works.
Some bard subclasses do have ways to expend bardic inspiration dice that don't narratively involve sound and whose rules don't mention sound or hearing1. They're a generalized resource for bards, like how some druid subclasses can "expend a use of wild shape" to do completely different things like summon wildfire spirit. So the basic way of expending them (inspiring an ally) doesn't have to work the same way as other ways of spending them.
In practice, the older Bard subclasses (published in the PHB: Lore and Valor) do only use their inspiration dice by doing audible things, but that was a conclusion reached by checking on their available uses, not an assumption I made to figure out how their uses worked.
To decide if a use of inspiration dice can work in silence, look for:
- Rules text that explicitly says no (or yes).
- Narrative flavour that definitely implies sound or words are involved. (If it makes sense and there's no rule against it, a player may come up with alternate flavour for their bard, different from some other bards.)
For Cutting Words, the last sentence of its rules paragraph is crystal clear:
The creature is immune if it can't hear you or if it's immune to being charmed.
So you can't be in Silence, and they can't be in Silence or Deafened. And there can't be a barrier to sound between you, or something like gale force winds or skydiving that drown out verbal communication.
If not for that, we could also infer from the name ("Cutting words") that it involves language. Unless the bard and the target were using sign language, or the bard was holding up a sign with written words, "words" implies spoken language. So a sensible DM ruling would be that a bard must be heard to weave their distracting magic. Other rulings would be possible, especially if a player had a plausible explanation involving gestures. But in this case there is explicit rules text saying that hearing is required, so DMs don't have to infer that from the narrative flavour.
Footnote 1: Uses of bardic dice that don't necessarily involve sound
(Xanathar's) Swords bard - blade flourishes. The plain English meaning of that is twirling your sword in your hand artistically, no implication of spoken language or sound. e.g. "Slashing Flourish. You can expend one use of your Bardic Inspiration to cause the weapon to deal extra damage to the target..." (Also, at 14th level, you can use a d6 on any flourish instead of expending an inspiration die.)
(Xanathar's) Whispers bard - Psychic Blades: "ability to make your weapon attacks magically toxic to a creature's mind." / When you hit a creature with a weapon attack, you can expend one use of your Bardic Inspiration to deal an extra 2d6 psychic damage...
The narrative explanation for this doesn't appear to involve speech or sound. Remember that the narrative flavour of a Whispers bard is that they hide in plain sight, often masquerading as members of another bard college. If they had to speak damaging words aloud to make that psychic damage happen, they'd reveal themselves.
By contrast, (Tasha's) College of Eloquence doesn't explicitly say the target must hear you for "Unsettling Words", but that's heavily implied.
You can spin words laced with magic that unsettle a creature and cause it to doubt itself. As a bonus action, you can expend one use of your Bardic Inspiration and choose one creature you can see within 60 feet of you. Roll the Bardic Inspiration die. The creature must subtract the number rolled from the next saving throw it makes before the start of your next turn.
Without explicit rules text about hearing, this leaves the door open to sign language if the target understands. It does say "words", though, so doing it with rude gestures would be a significant reflavouring if your party often uses Silence.
Another case where sound appears to be required is (Xanathar's) College of Glamour: Mantle of Inspiration
When you join the College of Glamour at 3rd level, you gain the ability to weave a song of fey magic that imbues your allies with vigor and speed.
As a bonus action, you can expend one use of your Bardic Inspiration to grant yourself a wondrous appearance. When you do so, choose a number of creatures you can see and that can see you within 60 feet of you, ...
The 2nd paragraph only mentions appearance, but the first paragraph says it's a song. If some of your allies are silenced/deaf but you the bard aren't, you could weave the song and transform your appearance, it would be up to the DM whether allies that could see but not hear you would get the temp HP and the movement reaction.