if you like iNat’s standard heatmap, then those map tiles are available via https://api.inaturalist.org/v1/heatmap/{z}/{x}/{y}.png.
if you’re using R for just visualization (not interactivity), you probably need to use tmap or leaflet to display one of iNat’s observation map tilesets over some sort of basemap.
UTFGrids are only needed if you want to be able to interactively click on an XYZ tile map to show additional data or provide other functionality. that’s more complicated to do, and i’m not sure if anyone has already written something to do this in R, although i bet it would be possible (though maybe not easy) to extend leaflet in R to to use these.
you can also use the data from UTFGrids to make interesting custom maps, but that’s also more complex to do, and you’d probably have to extend leaflet or tmap to do this, as well.
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if you don’t understand XYZ map tiles, you can read a little bit about them here: https://forum.inaturalist.org/t/in-pursuit-of-mappiness-part-1/21864#what-are-map-tiles-3.