CheckCollectionView.swift

import UIKit


class CheckCollectionView: UICollectionView {

    let reuseID = "cell"
    
    let checkColor = UIColor.blue   //チェックしたもののみ色をつける
    let defaultColor = UIColor.white
    
    //からの配列を用意
    var checkArray:[IndexPath] = []
    
    required init?(coder aDecoder: NSCoder) {
        super.init(coder: aDecoder)
    }
    
    override init(frame: CGRect, collectionViewLayout layout: UICollectionViewLayout) {
        super.init(frame: frame, collectionViewLayout: layout)
        self.delegate = self
        self.dataSource = self
        self.register(CustomCell.self, forCellWithReuseIdentifier: reuseID)
    }

}

extension CheckCollectionView: UICollectionViewDelegate {
    func collectionView(_ collectionView: UICollectionView, didSelectItemAt indexPath: IndexPath) {
        guard let cell:CustomCell = collectionView.cellForItem(at: indexPath) as? CustomCell else { return }
        
        //位置を探す
        if let index = checkArray.index(where: { (index) -> Bool in
            return index == indexPath
        }){
            //見つかったら配列から削除する
            checkArray.remove(at: index)
            cell.contentView.backgroundColor = defaultColor //セルの色をデフォルトに戻す
        }else {
            //配列内にないので追加する
            checkArray.append(indexPath)
            cell.contentView.backgroundColor = checkColor   //背景に色をつける
        }
    }
}

extension CheckCollectionView: UICollectionViewDataSource {
    func collectionView(_ collectionView: UICollectionView, numberOfItemsInSection section: Int) -> Int {
        return 100
    }
    
    func collectionView(_ collectionView: UICollectionView, cellForItemAt indexPath: IndexPath) -> UICollectionViewCell {
        let cell:CustomCell = collectionView.dequeueReusableCell(withReuseIdentifier: reuseID, for: indexPath) as! CustomCell
        
        cell.label.text = String(indexPath.row) //適当に行番号を表示しておく
        
        //チェックされているかどうかで色を分岐
        if checkArray.contains(indexPath) {
            cell.contentView.backgroundColor = checkColor
        }else {
            cell.contentView.backgroundColor = defaultColor
        }
        
        return cell
    }
}

CustomCell.swift

import UIKit

class CustomCell: UICollectionViewCell {
    
    var label:UILabel!

    required init?(coder aDecoder: NSCoder) {
        fatalError("init(coder:) has not been implemented")
    }
    
    override init(frame: CGRect) {
        super.init(frame: frame)
        //xibを使わない場合はこちらでセットアップ
        setup()
    }
    
    func setup() {
        label = UILabel(frame:self.contentView.frame)
        label.textAlignment = .center
        self.contentView.addSubview(label)
    }
    
}

ViewController.swift

import UIKit

class ViewController: UIViewController {

    override func viewDidLoad() {
        super.viewDidLoad()

        //適当にコレクションビューを設置
        let layout = UICollectionViewFlowLayout()
        layout.itemSize = CGSize(width:50,height:50)
        let collectionView = CollectionView(frame: self.view.frame, collectionViewLayout: layout)
        self.view.addSubview(collectionView)
    }

    override func didReceiveMemoryWarning() {
        super.didReceiveMemoryWarning()
        // Dispose of any resources that can be recreated.
    }

}