This depends on how many options you have for using soql. One way I can see it is:
Object__c myObject = new Object__c();
object.Name = 'A,B,D';
object.Multi_Select_Name__c = 'A;E';
insert object;
String query = 'select Id from Object__c';
//now we want to do a multi picklist, which is a String separated by ';'
List<String> nameSelections = object.Multi_Select_Name__c.split(';');
if(nameSelections.size() > 0) {
query += ' where ';
for(Integer i = 0; i < nameSelections.size(); i++) {
nameSelections[i] = 'Name like \'%' + nameSelections[i] + '%\'';
}
query += String.join(nameSelections, ' or ');
}
List<Object__c> objects = (List<Object__c)database.query(query);
This will create a query like:
'select Id from Object__c where Name like '%A%' or Name like '%E%'
Selecting all Object__c's where Name contains either A or E, generated from your multiselect.
Syntax errors may apply, I just wrote this real quick :) Ill double check myself.
EDIT:
This code works to build the query
String query = 'select Id from Object__c';
String multiSelect = 'A;E';
List<String> nameSelections = multiSelect.split(';');
if(nameSelections.size() > 0 ){
query += ' where ';
for(Integer i = 0; i < nameSelections.size(); i++) {
nameSelections[i] = 'Name like \'%' + nameSelections[i] + '%\'';
}
query += String.join(nameSelections, ' or ');
}
system.debug(LoggingLevel.INFO, query);