How to clear leaflet map of all markers and layers before adding new ones?

If you want to remove all the current layers (markers) in your group you can use the clearLayers method of L.markerClusterGroup(). Your reference is called markers so you would need to call:

markers.clearLayers();

You're losing the marker reference because it's set with var. Try saving the references to 'this' instead.

mapMarkers: [],
map: function (events) {
    [...]
    events.forEach(function (event) {
        [...]
        // create the marker
        var marker = L.marker([event.location.lat, event.location.lng]);
        [...]
        // Add marker to this.mapMarker for future reference
        this.mapMarkers.push(marker);
    });
    [...]
}

Then later when you need to remove the markers run:

for(var i = 0; i < this.mapMarkers.length; i++){
    this.map.removeLayer(this.mapMarkers[i]);
}

Alternatively, instead of saving each reference to each marker, you can just save the cluster to 'this'.