Looks like what are sometimes called cliff hanger shelving. The metal plates are from other fittings usually bathroom towel, toilet roll holders etc.
The brackets are screwed to the wall ( looks like previously to a plasterboard wall for which they are unsuitable.)
The the shelf is clamped between upper and lower plates, the half round pieces being covers for the lower clamp.
Similar type to these https://www.swanseatimber.co.uk/she...MIob_42Iic2gIV55PtCh1aTgCoEAQYFyABEgKS2PD_BwE
If you need to find a tradesperson to get your job done, please try our local search below,
or if you are doing it yourself you can find suppliers local to you.
Select the supplier or trade you require, enter your location to begin your search.
Please select a service and enter a location to continue...
Are you a trade or supplier? You can create your listing free at DIYnot Local