When a part anchors it flings me off?

I think it's because you are using :GetChildren() instead of :GetDescendants(). :GetChildren only gets whatever is directly under the object (in this case workspace), not all of that childs descendants.

In your case, whatever part you are trying to remove the velocity from could be under another child, so it wouldn't get set to 0.

Give this a shot and let me know if it worked!

Code:

local UIS = game:GetService("UserInputService")
UIS.InputBegan:Connect(function(input, typing)
    if typing then
        return end
    elseif input.KeyCode == Enum.KeyCode.Q then
        for i, v in pairs(game.Workspace:GetDescendants()) do
            if v:IsA('Part') or v:IsA('MeshPart') then
                if not v:FindFirstChild('Value') then
                    if v.Anchored == true then
                    	v.Anchored = false
                    elseif v.Anchored == false then
                        v.Anchored = true
                end
            end
        end
    end
end)

-- This goes in Starter Pack

p.s make sure to accept my answer if it helped!