You would do this by making requests to the twitter API.
There are lots of tutorials online for getting a tweet using the twitter API. To do this with Laravel you will likely want to use CURL, and there are literally 1000s of tutorials on how to do this using CURL.
Try searching for
curl php twitter api tutorial.
See how far you can get on your own, if you need help getting over a hurdle post back and I (or someone hopefully) will try to help you through it.
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