Qué alivio casi no se
ve la herida del pulgar

Why is 'se' needed here? I thought that 'se ver' meant 'look.'

Thank you

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It's not "se ver" but "verse" (yes, it's difficult)

You're right that "verse" can mean "to look (in a certain way)", like: Te ves cansado = You look tired
But we also say that something "se ve" to mean that it's visible (or that something "no se ve", to mean that it's not visible). For example:
No se ve nada = Literally: "nothing is seen", we'd use it as "I can't see anything"
Se ve humo a lo lejos = Literally: "smoke is seen in the distance", we'd use it as "I can see smoke in the distance"

As you can tell by the literal translations, "se" makes it impersonal, it's not that you can or can't see it because of your own vision or your good or bad eyesight, it's either visible or it isn't

So as for the sentence you showed:
"Qué alivio, casi no se ve la herida del pulgar"
It'd mean
"What a relief, the wound in the thumb is almost not visible"