The context for the prohibition of making graven images is within the greater prohibition of idolatry, and the emphasis of why they shouldn't make graven images is given at the end of the verse you cited; you shouldn't bow down or serve idols.
Exodus 20:4-5 says, "Thou shall not make unto thee any graven image,
or any likeness of any thing that is in heaven above, or that is in
the earth beneath, or that is in the water under the earth: Thou shalt
not bow down thyself to them, nor serve them."
Reading the rest of the chapter for context sheds further light on what this prohibition really means, specifically in verse 23:
22 The Lord said to Moses, “Thus you will tell the
Israelites: ‘You yourselves have seen that I have spoken with you from
heaven. 23 You must not make gods of silver alongside me, nor make
gods of gold for yourselves. Exodus 20:22-23 NET
When all of these verses are taken together, not as disparate comments but as further instructions expanding upon one topic, namely, that of idolatry, then the understanding is that the prohibition against making graven images is specifically against images made for the purposes of idolatry.
Put in other words, the Israelites are prohibited from making anything with the express purpose of worshipping it; or making anything to be a substitute for God ('you must not make gods of silver alongside me' as verse 23 says).
To quote the Talmudic commentary on Exodus 20:4-5 and the prohibition of making graven images:
כי תניא ההיא לעובדם When that baraita is taught, it is in reference
to the prohibition against worshipping them. There is no prohibition
against forming a figure in their likeness.
Talmud Avodah Zarah 43b7
Finally, as proof that this prohibition was not against 'art' or 'images' in of themselves, we can look only 5 chapters further in Exodus, when God commands them to make cherubim angles for the arc:
18 You are to make two cherubim[a] of gold; you are to make them of
hammered metal on the two ends of the atonement lid. 19 Make[b] one
cherub on one end[c] and one cherub on the other end; from the
atonement lid[d] you are to make the cherubim on the two ends. 20 The
cherubim are to be spreading their wings upward, overshadowing[e] the
atonement lid with their wings, and the cherubim are to face each
other,[f] looking[g] toward the atonement lid. Exodus 25:18-20 NET
In summary, the greater context of the passage teaches that the fabrication of anything for the express purpose of worshipping it; no matter what it is, is prohibited, because God does not want us to substitute other things in his place.
Graven images can be made, but only if their purpose is not for idolatry, nor to replace God.