Michael Graf already pointed out the basic way to get it working, so here a few additional information bits about the LAPC-I that may help to understand the setup involved:
Name
- It's mostly known that
LA
stands for Linear Arithmetic (Synthesizer)
- Unlike assumed
PC
stands not for Personal Computer but
P
complements LA
as Processor, while
C
simply stands for Card.
- Similar is
I
not not the Roman numeral for one but the letter 'I', for IBM-PC.
- Thus
LAPC-I
means Linear Arithmetic Processor Card for the IBM-PC
Whats In It?
A LAPC-I is essentially the combination of a
- Roland MT32 Multi-Timbral sound generator
- Roland MPU-IPC-T - MIDI Processor Unit for the IBM PC - which is a combination of the prior
- Roland MIF-IPC - MIDI Inter Face for the IBM PC - with all of the
- Roland MPU-401 breakout box except for the connectors
MIF cards were available for many common computer, especially Japanese systems, but also C64 or Apple II. They 'only' provided the MIDI interface, all further electronics were part of the MPU-401 box. The MPU-IPC-T moved much of onto it's card reducing the break out box size.
The MT32 in turn was the syntesizer to be controlled via MIDI. It was considered a 'low end' offering for 'Hobbyist' musicians.
The LAPC-I integrated all of that onto one board needing only the smaller break out box (now called MCB-1 - MIDI Connector Box), as seen in above picture, to support connection with other MIDI devices (Manual).
Why Does It Work Without PC
The MT32 is a stand alone synthesizer controlled over MIDI. It does not need a PC to work. The PC's MIDI interface part will when only powered be strictly invisible to the bus - as shown in the manual:

(From the LAPC-I Manual p.24)
All input at MIDI-IN will be handed to MIDI-OUT and thus the MT32 part. For all that matters it will look and work like any stand alone MT-32.
How To Connect
The LAPC-I became famous for high end gamer setup - and that created one problem for today's usage: Gamers almost never bought the MCB1, which was sold separat. Thus the MCB-1 is a rather rare sight.
But since the connector used (DA-15) is a common one and the MCB-1 insides known a rebuild shouldn't be too hard.

Alternative Solutions
Get Yourself a Standard MT-32 Box.
While 'hacking' a LAPC-I to work as a stand alone MT-32 is nice, it involves a lot of work and at least basic electronic skills if no MCB-1 is at hand. In contrast any MT-32 will already include the necessary connectors, reducing the task to getting an USB to MIDI interface and plug it into PC and MT-32. No ISA setup, no external power supply, no soldering, just plug and play :))
Also, MT-32 are way more common and available at way lower prices than a LAPC-I.
Still Insisting to "Hack" it?
Well, as a quick look at some of my LAPC-I

shows, it connects only
- GND - at B1 / B10 / B31
- +5V - at B3 / B29
- -5V - at B5
for power. Thus a simple 62 Pin Card Edge Connector at ca. USD 2, a +/-5V PSU of your choice (*1) and 6 wires connecting those two will do it.
While it's really a little fun job, I would still suggest to be careful and count everything twice - at least.
*1 - Quickhack would be to chain two cheap USB wall warts.