Tamsulosin has two modes of working.
Firstly, it relaxes the muscles of the prostate and internal urinary sphincter, which should cause you to pee faster. This effect is pretty instant, and also wears off within a day (typically within 16 hours actually).
Secondly, long term use is sometimes claimed to shrink the prostate, although this effect is sufficiently minor than many urologists claim it's not actually true.
It's worth explaining that peeing slowly can result in not emptying your bladder before you finish peeing, because you pee for a length of time, and not until your bladder is empty. If you are peeing slowly, you will finish peeing while still having urine in your bladder, so you'll want to pee again sooner. This residual left in the bladder is the most common cause of UTIs in men.
Also peeing slowly can result in getting an unstable bladder, where you get the urgency to pee even though your bladder isn't full. If the Tamsulosin fixes the slow flow rate, then your unstable bladder may resolve some months later, but doesn't always. There are some medication available to help with this, such as Solifenacin, Mirabegron, Trospium Chloride. These may increase the risk of not emptying your bladder though.