I understand that the cultures and languages of those nations were already firmly rooted, but that was pretty similar to Algeria and Morocco who still ended up using French (and in Western Sahara, Spanish) quite extensively to this day. Why was this not the case in the Italian colonies?
I'm just going to respond regarding Somalia, since I'm not as well read regarding Libya and other Italian colonial territories.
The short answer to your question is that Italy's footprint in Somalia was light for most of its colonial history, and just as the Italians were ramping up assimilation efforts, WWII came along and stripped the colony from them.
The area became a protectorate of Italy in 1888 when Sultan Yusuf Ali Kenadid of the Majeerteen Sultanate signed a treaty with Italy. Kenadid's rival and neighbor, Sultan Boqor Osman Mahamuud of the Sultanate of Hobyo, signed a similar deal the next year, forming a continuous territory.
It was not, in some ways, a traditional colony at first in that the Italians had limited interest in most of the arid Somalian landscape - they were mostly interested in the strategically important ports on the shore, which gave them access to the Suez Canal and the Gulf of Aden.
The Italians provided arms and money to the sultans in return for this protectorate status, but they held little direct control beyond their coastal strip. Even here, the Italian population was mostly military officers and colonial officials, with few settlers.
It wasn't until the early 1930s that a two successive Italian Governors - Guido Corni and Maurizio Rava - started actively trying to assimilate the Somalis into Italian culture. The Italians began to build out hospitals and schools around the capital, Mogadishu, around this time, and more Italians began to arrive as permanent settlers. By 1930, there were 22,000 Italians living in Mogadishu and its immediate surrounding territory, but they were still a light footprint on the ground.
By 1935 that number had grown to 50,000 Italians settlers, 20,000 of who lived in Mogadishu, representing just under half of the city's population. Over the course of the colonial governance of the territory, the succession of sultans which controlled much of the inland territory occasionally proved disloyal, and over time the Italians started taking more and more direct control of the hinterlands of the territory as a way of ensuring better control.
If things had continued like this for a few more decades it's very likely that Italian language and culture would have become inculcated, at least around the coastal areas where Italian presence was heaviest, but Italy joined the Axis during WWII. They lost Somalia to a British attack in the spring of 1941. After the war the UN made the colony a trusteeship, and it was granted independence in the 1960s.
When Siad Barre became president of Somalia in 1969, he enforced a policy of promoting the Somali language and enforcing it in government use. He adopted a new Latin alphabet for the language. Somali was the only language allowed to be taught in schools. His stated reason was to erase the gap between those who fluently spoke Italian and English and those who did not, as well as to help foster a thriving Somali culture.
As to the Italian colonists, some started leaving after WWII, but the exodus picked up after independence. There were several anti-Italian riots in Mogadishu around this time in which multiple Italians were killed. Many returned to Italy or migrated to the US, though there was still a small Italian population in the city through the 1990s.
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