How can I remove the background / button from the homepage slider?

Remove grey button & box from carousel, or remove the grey background.

Open up customise.css in your template editor (Themes > Template Editor > customise.css.  There are three options here:

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Option #1: If you *don't* want to keep the text,

i.e. if you want to remove the grey background and caption text and button, add this to customise.css:

.carousel-caption {display:none;}

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Option #2: If you want to keep the text, but only remove the background & button, add this instead:

#slider.carousel div.carousel-inner div.item div.container div.carousel-caption a.btn {display:none;}
.carousel-caption { background-color: transparent;}

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Option #3: If you want to only remove the background, add this instead:
.carousel-caption { background-color: transparent;}

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And you're done.