Is it safe to use frayed MagSafe with outer insulation part is broken
Will it damage the logic board?

No.  The shielding is also the ground conductor.  So, you need it for two reasons.  If you cut it, you’ll be breaking the MacBook’s connection to ground which of course will affect it’s ability to function.  It’s like disconnecting the negative battery terminal from your car or any other battery power source - it will fail to function.

Will the &quot;cracked&quot; wire of a MacBook charger eletrocute me

It’s not “dangerous” to you, but if you were to somehow send voltage down this line you could damage your charger and/or Mac.  However, those vintage Mac laptops had a separate (replaceable) DC-in board that would take the brunt of any surge.  If there was damage, it would be contained there.