According to supply and demand, when there’s an increase in supply, resulting in a surplus, the price falls till it reaches equilibrium. But how exactly does this work? Let’s say the equilibrium is 200 widgets at 100 dollars each. I produce 300 widgets to sell them at 150 dollars each, but the problem is people will only demand 100 widgets at 150 dollars each. This will cause a surplus. Am I supposed to lower the price of each widget to 100 dollars (the equilibrium price)? If so, I will only be able to sell 200 of my 300 widgets, leaving 100 widgets left over. What do I do with the remaining supply?
It's your choice. You could dump them in the landfill. Economists generally assume that you won't dump them in the landfill, and you would rather sell them for some money than no money, therefore you will lower the price until you sold them all.
If that means lowering the price below what they cost to make, then you made too many and next week you will make less. Or (more likely) you will try to balance it out by storing them in a warehouse and selling them next week instead of this week, which is equivalent to producing less this week and producing more next week.
And if many competing firms do these sensible things, the price will end up being the equilibrium price. If there isn't sufficient competition, a monopoly firm may find it more profitable to reduce supply by dumping its product in a landfill or store them in a warehouse indefinitely. Then the price will end up being the profit-maximizing price for that firm.
You can't just make up arbitrary equilibrium price at the same time as you are making arbitrary supply and demand. Equilibrium price is a price at which supply equals demand. What you are saying is equivalent to saying if $d=rt$ (distance equals rate time time), and I assume rate is 10 km per hour and time 1 hour, how come I have travelled 20km? Well the answer is that you simply concocted impossible scenario, either you cannot travel at that speed or you could not traverse 20km or you had to travel more than 1 hour.
Similarly you can't just make up equilibrium price. Equilibrium price will be the price $p^*$ at which $D(p)=S(p)$. For example, if demand is $D=100-p$ and supply is $S=10+p$ then equilibrium price is $p^*=45$. You can't just say lets say the equilibrium price is 100 (or to be more precise you could say that but in that case you can't also arbitrarily select some demand and supply not consistent with equilibrium price).
Hence you cannot say that equilibrium is 200 widgets at 100 dollars each.
If so, I will only be able to sell 200 of my 300 widgets, leaving 100 widgets left over. What do I do with the remaining supply?
In the case you would either have to reduce price (until you reach actual equilibrium price) to sell all 300 widgets, or just produce less widgets (change supply).
What do I do with the remaining supply? If, at this price (\$100) the supply is still greater than the demand, you could sell more units by further decreasing the price. This would nicely answer your question of "How does price decrease when supply increases?"
Some remarks: "when supply increases" is quite multifaceted. In this context it would probably mean that the supply curve has tilted/shifted downward due to a new entrant/technological improvement.
The perfect competition model assumes agents without market power; that is, each firm is so small ("atomic") that were they to price above the market price none of their goods would be bought, and they have no effect on the market price.
Thus, a more accurate statement, is that within the perfect competition model, if the supply curve shifts/tilts so that supply increases, the new equilibrium price will be lower than the old one (if demand is strictly decreasing in price).