Storm is of course related to the English word 'storm' and refers to a period of bad weather with strong winds; under the meteorological definition, wind speed must be above 89 kilometres per hour to be classed as stormy (or stormig in Swedish) but in practice the word is also used in other contexts. It's a very old word, also linked to German Sturm, Gaelic and Irish stoirm, and Icelandic stormor, all of which come from a proto-Germanic word sturmaz.
In Swedish, you can also turn storm into a verb: storma, as in det stormar (literally 'it is storming', but in English you could only say 'there is a storm' or 'the weather is stormy').
By has several meanings in Swedish, and the one that most language learners are likely to learn first is 'village', but that's not the right one for stormby. By meaning a small village is related to the earlier word býr (town or farm), and shares its origins with the verb bo (to live or reside) and noun bygd (settlement).
But by can also mean 'gust' or 'blast', and Swedish also has the adjective byig meaning 'windy/gusty'. So stormby means 'stormy winds/gusts', and you may also hear or read the linked words kastby or vindby.
Each one has a slightly different precise meaning that can help you determine the strength of a storm. Vindby means 'squall' or 'gale' and refers to a sudden and usually short-term increase in wind speed, while stormby is slightly more intense and means that occasional squalls reach the level classed as a storm, although the average wind speed is lower than that classed a storm. Finally, kastby refers to sharp changes in the wind's direction.
Since gusts are rarely a one-off event, you'll almost always see these three words in the plural form: stormbyar, vindbyar, and kastbyar.
Från kvällen väntas stormbyar över hela Gotland
From the evening, stormy gusts are expected over all of Gotland
SMHI varnar för risk för nordliga stormbyar
SMHI warns of a risk of stormy gusts in the north