Source: ApexLog

The ApexLog gives access to the Debug Log.

To enable log monitoring for a user, go to Setup > Logs > Debug Logs and click the New button to add an user.

These rows can be queried:

select id, StartTime, Application, DurationMilliseconds, Location, LogLength, LogUser.Name, Operation, Status from apexlog order by StartTime DESC

You can read all the infos regarding the given apex log, such as the User, kind of request, the size of the log, its status, but you cannot access the raw content: to get to the raw content you have to make a Tooling API request to the endpoint:

GET https://xxx.salesforce.com/services/data/v34.0/sobjects/ApexLog/[APEXLOG_ID]/Body/
 Headers:
  Authorization: Bearer [SESSION_TOKEN]

The session token could be grabbed with an anonymous Apex call to UserInfo.getSessionId() or using the OAuth Password flow described in the ActionLinkGroupTemplate object.

This is what you get:

34.0 APEX_CODE,FINEST;APEX_PROFILING,INFO;CALLOUT,INFO;DB,INFO;SYSTEM,DEBUG;VALIDATION,INFO;VISUALFORCE,INFO;WORKFLOW,INFO
17:13:22.020 (20181893)|EXECUTION_STARTED
17:13:22.020 (20210501)|CODE_UNIT_STARTED|[EXTERNAL]|06624000003BOE1|VF: /apex/ConfirmationTokenRequired
17:35:30.036 (36842637)|CODE_UNIT_FINISHED|execute_anonymous_apex
17:35:30.038 (38198449)|EXECUTION_FINISHED
17:13:22.040 (40498571)|CODE_UNIT_STARTED|[EXTERNAL]|ApexDataSource:MongoDBDataSrouceProvider
17:13:22.040 (40523619)|CODE_UNIT_STARTED|[EXTERNAL]|ApexDataSource:MongoDBDataSrouceProvider
17:13:22.043 (43448400)|HEAP_ALLOCATE|[71]|Bytes:3
17:13:22.043 (43497592)|HEAP_ALLOCATE|[76]|Bytes:152
17:13:22.043 (43514859)|HEAP_ALLOCATE|[272]|Bytes:408
17:13:22.043 (43534730)|HEAP_ALLOCATE|[285]|Bytes:408
17:13:22.043 (43553779)|HEAP_ALLOCATE|[379]|Bytes:48
17:13:22.043 (43582592)|HEAP_ALLOCATE|[131]|Bytes:6
17:13:22.043 (43605101)|HEAP_ALLOCATE|[EXTERNAL]|Bytes:6
17:13:22.043 (43624664)|METHOD_ENTRY|[1]|01p2400000AgxCj|MongoDBDataSrouceProvider.MongoDBDataSrouceProvider()
17:13:22.043 (43630791)|STATEMENT_EXECUTE|[1]
17:13:22.043 (43636403)|STATEMENT_EXECUTE|[1]
17:13:22.043 (43641248)|METHOD_EXIT|[1]|MongoDBDataSrouceProvider
17:13:22.043 (43660705)|HEAP_ALLOCATE|[EXTERNAL]|Bytes:4
17:13:22.043 (43671735)|VARIABLE_SCOPE_BEGIN|[1]|this|MongoDBDataSrouceProvider|true|false
17:13:22.043 (43713117)|VARIABLE_ASSIGNMENT|[1]|this|{}|0x74896a31
17:13:22.043 (43758665)|SYSTEM_CONSTRUCTOR_ENTRY|[EXTERNAL]|()
17:13:22.043 (43866237)|SYSTEM_CONSTRUCTOR_EXIT|[EXTERNAL]|()
17:13:22.043 (43880333)|STATEMENT_EXECUTE|[1]
17:13:22.043 (43892018)|CODE_UNIT_FINISHED|ApexDataSource:MongoDBDataSrouceProvider
17:13:22.044 (44048812)|HEAP_ALLOCATE|[EXTERNAL]|Bytes:4
17:13:22.044 (44059804)|SYSTEM_MODE_ENTER|false
17:13:22.044 (44067540)|VARIABLE_SCOPE_BEGIN|[7]|this|MongoDBDataSrouceProvider|true|false
17:13:22.044 (44087738)|VARIABLE_ASSIGNMENT|[7]|this|{}|0x74896a31
17:13:22.044 (44101508)|STATEMENT_EXECUTE|[7]
17:13:22.044 (44104497)|STATEMENT_EXECUTE|[8]
17:13:22.044 (44112485)|HEAP_ALLOCATE|[8]|Bytes:4
17:13:22.044 (44182135)|SYSTEM_CONSTRUCTOR_ENTRY|[8]|()
17:13:22.044 (44204845)|SYSTEM_CONSTRUCTOR_EXIT|[8]|()
17:13:22.044 (44210268)|VARIABLE_SCOPE_BEGIN|[8]|capabilities|List|true|false
17:13:22.044 (44237205)|HEAP_ALLOCATE|[EXTERNAL]|Bytes:4
17:13:22.044 (44259684)|VARIABLE_ASSIGNMENT|[8]|capabilities|{"s":1,"v":[]}|0x3ce93902
17:13:22.044 (44264466)|STATEMENT_EXECUTE|[9]
17:13:22.044 (44381336)|SYSTEM_METHOD_ENTRY|[9]|List.add(Object)
17:13:22.044 (44408889)|HEAP_ALLOCATE|[EXTERNAL]|Bytes:4
17:13:22.044 (44420907)|SYSTEM_METHOD_EXIT|[9]|List.add(Object)
17:13:22.044 (44426463)|STATEMENT_EXECUTE|[10]
17:13:22.044 (44439376)|SYSTEM_METHOD_ENTRY|[10]|List.add(Object)
17:13:22.044 (44453626)|HEAP_ALLOCATE|[EXTERNAL]|Bytes:4
17:13:22.044 (44462615)|SYSTEM_METHOD_EXIT|[10]|List.add(Object)
17:13:22.044 (44466873)|STATEMENT_EXECUTE|[11]
17:13:22.044 (44476051)|SYSTEM_MODE_EXIT|false
17:13:22.044 (44490717)|CODE_UNIT_FINISHED|ApexDataSource:MongoDBDataSrouceProvider
17:13:22.061 (61085594)|CUMULATIVE_LIMIT_USAGE
17:13:22.061 (61085594)|LIMIT_USAGE_FOR_NS|(default)|