According to Sugar.org, this is how sugar is made from either beets or cane.
For sugar cane:
❧ Grinding the cane to extract the juice;
❧ Boiling the juice until the syrup thickens and crystallizes;
❧ Spinning the crystals in a centrifuge to produce raw sugar;
❧ Shipping the raw sugar to a reﬁnery where it is,
❧ Washed and ﬁltered to remove remaining non-sugar ingredients and color; and
❧ Crystallized, dried and packaged.
Beet sugar processing is similar, but it is done in one continuous process without the raw sugar stage. The sugar beets are washed, sliced and soaked in hot water to separate the sugar-containing juice from the beet ﬁber. The sugar-laden juice is puriﬁed, ﬁltered, concentrated and dried in a series of steps similar to sugar cane processing.
The below illustration I took from the pamphlet on Sugar.org called "How Well Do You Know Sugar?" located at this link.