The numbers and letters [4,3-fg] indicate the side where the fusion occurs and the relative orientation of the fused ring system. The direction of the numbers of the attached component (e.g. ‘3,4’ or ‘4,3’) corresponds to the direction of the lettering of the peripheral sides of the parent component (i.e. a, b, c, etc.).
The correspondig Subsection P-25.3.1.3 in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), reads as follows:
(…)
Isomers are distinguished by lettering, continuously, each peripheral side of the parent component (including sides whose locants are distinguished by letters, for
example, 2a,3a) using the italic letters a, b, c, etc., beginning with a for the side numbered ‘1,2’, b for ‘2,3’ etc. To the letter as early in the alphabet as possible that denotes the side where the fusion occurs are prefixed, if necessary, the numbers of the positions of attachment of the other component. These numbers are chosen to be as low as is consistent with the numbering of the compound and their order conforms to the direction of lettering of the parent component. (…)
For the compound that is given in the question, the fusion occurs at the side of quinoline between the atoms 5 and 6, which corresponds to the peripheral side f, and also at the side between the atoms 6 and 7, which corresponds to the peripheral side g. Thus, the parent component is described using the name [fg]quinoline.
The attached component is indole. The fusion occurs between the atoms 3 and 3a and also between the atoms 3a and 4. Now there are two possibilities for the relative orientation of the fused ring system indolo[3,4-fg]quinoline as well as indolo[4,3-fg]quinoline.
The direction of the numbers of the attached indole (e.g. ‘3,4’ or ‘4,3’) corresponds to the direction of the lettering of the parent quinoline (i.e. a, b, c, d, e, f, g, etc.). For the compound that is given in the question, the direction is from 4 to 3. Therefore, the name of the fused ring system is indolo[4,3-fg]quinoline.
The fused ring system gets a new numbering of locants. Numbering is quite easy if you draw the structure correctly according to the IUPAC recommendations for orientation of fused ring systems (wich are beyond the scope of this question; see Subsection P-25.3.2.3 for details).
P-25.3.3.1 Numbering of peripheral skeletal atoms
P-25.3.3.1.1 The numbering of peripheral atoms in the preferred orientation starts from the uppermost ring. If there is more than one uppermost ring, the ring furthest to the right is chosen. Numbering starts from the nonfused atom most counterclockwise in the ring selected and proceeds in a clockwise direction around the system, including fusion heteroatoms but not fusion carbon atoms. Each fusion carbon atom is given the same number as the immediately preceding nonfusion skeletal atom, modified by a Roman letter ‘a’, ‘b’, ‘c’, ‘d’, etc.
When the numbering of a system is fixed like that, this numbering must be used. Suffixes and prefixes are added to the name, as required, using the new locants characterizing the fused system. In particular, the saturation of the compound that is given in the question is indicated using hydro prefixes as 4,6,6a,7,8,9-hexahydroindolo[4,3-fg]quinoline.
The principal characteristic group is expressed as suffix ‘carboxamide’.
The remaining substituent groups are added as prefixes (‘methyl’ and ‘ethyl’).
Finally, stereodescriptors are placed at the front of the complete name or name fragment to which they apply. So the complete name is (6aR,9R)-N,N-diethyl-7-methyl-4,6,6a,7,8,9-hexahydroindolo[4,3-fg]quinoline-9-carboxamide.