I have seen commentary explaining that since the Bnai Yisrael could conquer and destroy the surrounding nations, Moav would lose all of its trading partners. Thus, while leaving Moav alone, they would lick up the source of wealth that Moav gained from all of its neighbors, leaving it isolated in what was now a barren and worthless area. An ox uses its tongue to rip the grass out of the ground in order to eat it, not even leaving the roots. The analogy is to oxen totally destroying a grazing area as they rip the grass away from the ground and leaving it barren.
Note that the message sent to the elders of Midyan said that the entire surroundings of Moav (all the neighboring nations) would be destroyed.
As the pasuk says:
Moab said to the elders of Midian, "Now this assembly will eat up
everything around us, as the ox eats up the greens of the field. Balak
the son of Zippor was king of Moab at that time.
Rashi on Balak 22:4 explains this as:
as the ox eats up: Whatever the ox has eaten up no longer contains
blessing [because the ox uproots the plants it eats (Da’ath Zekenim)].
— [Mid. Tanchuma Balak 3, Num. Rabbah 20:4]
אפילו שלא ילכדו את ארצנו ילחכו ברובם את כל סביבותינו, כלחוך השור את
ירק השדה, וילכדו להם את כל סביבותינו כאשר עשו לשני מלכי האמורי ויתנו
אותנו למס עובד.
Even if they do not conquer our land, they will lick up in their
numbers all of our surroundings, like the ox licks up the greenery of
the field, and they will conquer all of our surroundings as they did
to the two Amorite kings and they will take us as tax slaves.
The אור החיים explains that this implies the entire surrounding area and not just Moav
ילחכו וגו׳ את כל סביבותינו – אולי שלא רצו להראות כל כך מורך הלבב שהם
יראים מהם אלא שהם חסים על הסביבות, ואמרו לשון רבים לכלול אותם עמהם,
It appears that they did not want to express such fear on themselves,
but that they were worried on the surroundings, and they expressed
themselves in the plural language to include the others with them.
While Bnai Yisrael had promised not to harm anyone who allowed it to pass through, they were intent on conquering the land of Canaan. This would have destroyed the major economic strengths of the area. The Egyptians had already lost their economic power as a result of the makos. Once the Bnai Yisrael had conquered Canaan, the trade routes to Egypt would be cut off and trade would not resume once Egypt had recovered (and more than a full generation had passed so it may already have done so). Also the fact of what they had done to Sichon when he attacked them showed what they could do. An analogy could be made to the Great Depression of 1929 which built up as a result to the aftermath of World War I. Even the winning nations of Europe were in major economic trouble.
Rav Hirsch shows that לחך implies total consumption.
לחך means to lick, to lick up; to lick up water or dust with the
As he explains
Now as a matter of fact the ox first grasps with its tongue, the grass
which it tears up and swallows. So that one can literally say "the ox
licks up the grass", and לחך here is an apt expression. But the sense
of the message was: - as naturally and effortless as the ox takes the
grass for its food, so naturally and easily will we all become the
booty of this קהל.